A cube is painted blue on all its faces and then cut into 216 equal smaller cubes of identical size. How many of these smaller cu…

Question

A cube is painted blue on all its faces and then cut into 216 equal smaller cubes of identical size.

How many of these smaller cubes have exactly one face painted blue?

1. A. 36
2. B. 72
3. C. 100
4. D. 96 (Correct answer)

Correct Answer

Option D — 96

Detailed Solution & Explanation

The correct answer is 96.

Key Points

• The cube is cut into 216 = 6³ smaller cubes, so each edge is divided into 6.
• A small cube has exactly one painted face when it lies in the interior of a face — not touching any edge of the big cube.
• On each face that interior block measures (6 − 2) × (6 − 2) = 4 × 4 = 16 cubes.
• With 6 faces: 6 × 16 = 96.

Additional Information

• The general formulae for an n × n × n painted cube are: exactly one face painted = 6(n − 2)²; exactly two faces = 12(n − 2); exactly three faces = 8 (the corners); no face painted = (n − 2)³.
• For n = 6 these give 96, 48, 8 and 64, which correctly sum to 216.

*Ministry of Papers worked solution. The Commission has not yet published the official key for this paper; this answer will be reconciled with it on release.*